Optimal Estimation of Dynamic Systems, 2nd Edition

330 Optimal Estimation of Dynamic Systems

Equation (5.21) gives the desired form for the smoothed state estimate using the

combined forward and backward state estimates.

With the definitions of P

−

bk

and P

+

bk

, the inverse of the backward update covari-

ance follows directly from the information filter of §3.3.3, given by Equation (3.78):

P

+

bk

= P

−

bk

+ H

T

k

R

−1

k

H

k

(5.23)

To derive a backward recursion for P

−

bk

we first subtract Equation (5.3) from Equa-

tion (5.4), and use the error definitions

˜

x

−

bk

=

ˆ

x

−

bk

−x

k

and

˜

x

+

bk

=

ˆ

x

+

bk

−x

k

to give

˜

x

−

bk

= Φ

−1

k

˜

x

+

bk+1

+ Φ

−1

k

ϒ

k

w

k

(5.24)

Since

˜

x

+

bk

and w

k

are uncorrelated, then applying the definition in Equation (5.9c)

with Equation (5.24) ...